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Theorem 2spthonot 24542
Description: The set of simple paths of length 2 between two vertices (in a graph) as ordered triple. (Contributed by Alexander van der Vekens, 1-Mar-2018.)
Assertion
Ref Expression
2spthonot  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  ( A  e.  V  /\  B  e.  V ) )  -> 
( A ( V 2SPathOnOt  E ) B )  =  { t  e.  ( ( V  X.  V )  X.  V
)  |  E. f E. p ( f ( A ( V SPathOn  E
) B ) p  /\  ( # `  f
)  =  2  /\  ( ( 1st `  ( 1st `  t ) )  =  A  /\  ( 2nd `  ( 1st `  t
) )  =  ( p `  1 )  /\  ( 2nd `  t
)  =  B ) ) } )
Distinct variable groups:    t, E, f, p    t, V, f, p    A, f, p, t    B, f, p, t
Allowed substitution hints:    X( t, f, p)    Y( t, f, p)

Proof of Theorem 2spthonot
Dummy variables  a 
b are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 is2spthonot 24540 . . . 4  |-  ( ( V  e.  X  /\  E  e.  Y )  ->  ( V 2SPathOnOt  E )  =  ( a  e.  V ,  b  e.  V  |->  { t  e.  ( ( V  X.  V )  X.  V
)  |  E. f E. p ( f ( a ( V SPathOn  E
) b ) p  /\  ( # `  f
)  =  2  /\  ( ( 1st `  ( 1st `  t ) )  =  a  /\  ( 2nd `  ( 1st `  t
) )  =  ( p `  1 )  /\  ( 2nd `  t
)  =  b ) ) } ) )
21adantr 465 . . 3  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  ( A  e.  V  /\  B  e.  V ) )  -> 
( V 2SPathOnOt  E )  =  ( a  e.  V ,  b  e.  V  |->  { t  e.  ( ( V  X.  V
)  X.  V )  |  E. f E. p ( f ( a ( V SPathOn  E
) b ) p  /\  ( # `  f
)  =  2  /\  ( ( 1st `  ( 1st `  t ) )  =  a  /\  ( 2nd `  ( 1st `  t
) )  =  ( p `  1 )  /\  ( 2nd `  t
)  =  b ) ) } ) )
32oveqd 6299 . 2  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  ( A  e.  V  /\  B  e.  V ) )  -> 
( A ( V 2SPathOnOt  E ) B )  =  ( A ( a  e.  V , 
b  e.  V  |->  { t  e.  ( ( V  X.  V )  X.  V )  |  E. f E. p
( f ( a ( V SPathOn  E )
b ) p  /\  ( # `  f )  =  2  /\  (
( 1st `  ( 1st `  t ) )  =  a  /\  ( 2nd `  ( 1st `  t
) )  =  ( p `  1 )  /\  ( 2nd `  t
)  =  b ) ) } ) B ) )
4 simprl 755 . . 3  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  ( A  e.  V  /\  B  e.  V ) )  ->  A  e.  V )
5 simprr 756 . . 3  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  ( A  e.  V  /\  B  e.  V ) )  ->  B  e.  V )
6 xpexg 6709 . . . . . . 7  |-  ( ( V  e.  X  /\  V  e.  X )  ->  ( V  X.  V
)  e.  _V )
76anidms 645 . . . . . 6  |-  ( V  e.  X  ->  ( V  X.  V )  e. 
_V )
8 xpexg 6709 . . . . . 6  |-  ( ( ( V  X.  V
)  e.  _V  /\  V  e.  X )  ->  ( ( V  X.  V )  X.  V
)  e.  _V )
97, 8mpancom 669 . . . . 5  |-  ( V  e.  X  ->  (
( V  X.  V
)  X.  V )  e.  _V )
109ad2antrr 725 . . . 4  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  ( A  e.  V  /\  B  e.  V ) )  -> 
( ( V  X.  V )  X.  V
)  e.  _V )
11 rabexg 4597 . . . 4  |-  ( ( ( V  X.  V
)  X.  V )  e.  _V  ->  { t  e.  ( ( V  X.  V )  X.  V )  |  E. f E. p ( f ( A ( V SPathOn  E ) B ) p  /\  ( # `  f )  =  2  /\  ( ( 1st `  ( 1st `  t
) )  =  A  /\  ( 2nd `  ( 1st `  t ) )  =  ( p ` 
1 )  /\  ( 2nd `  t )  =  B ) ) }  e.  _V )
1210, 11syl 16 . . 3  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  ( A  e.  V  /\  B  e.  V ) )  ->  { t  e.  ( ( V  X.  V
)  X.  V )  |  E. f E. p ( f ( A ( V SPathOn  E
) B ) p  /\  ( # `  f
)  =  2  /\  ( ( 1st `  ( 1st `  t ) )  =  A  /\  ( 2nd `  ( 1st `  t
) )  =  ( p `  1 )  /\  ( 2nd `  t
)  =  B ) ) }  e.  _V )
13 oveq12 6291 . . . . . . . 8  |-  ( ( a  =  A  /\  b  =  B )  ->  ( a ( V SPathOn  E ) b )  =  ( A ( V SPathOn  E ) B ) )
1413breqd 4458 . . . . . . 7  |-  ( ( a  =  A  /\  b  =  B )  ->  ( f ( a ( V SPathOn  E )
b ) p  <->  f ( A ( V SPathOn  E
) B ) p ) )
15 eqeq2 2482 . . . . . . . . 9  |-  ( a  =  A  ->  (
( 1st `  ( 1st `  t ) )  =  a  <->  ( 1st `  ( 1st `  t
) )  =  A ) )
1615adantr 465 . . . . . . . 8  |-  ( ( a  =  A  /\  b  =  B )  ->  ( ( 1st `  ( 1st `  t ) )  =  a  <->  ( 1st `  ( 1st `  t
) )  =  A ) )
17 eqeq2 2482 . . . . . . . . 9  |-  ( b  =  B  ->  (
( 2nd `  t
)  =  b  <->  ( 2nd `  t )  =  B ) )
1817adantl 466 . . . . . . . 8  |-  ( ( a  =  A  /\  b  =  B )  ->  ( ( 2nd `  t
)  =  b  <->  ( 2nd `  t )  =  B ) )
1916, 183anbi13d 1301 . . . . . . 7  |-  ( ( a  =  A  /\  b  =  B )  ->  ( ( ( 1st `  ( 1st `  t
) )  =  a  /\  ( 2nd `  ( 1st `  t ) )  =  ( p ` 
1 )  /\  ( 2nd `  t )  =  b )  <->  ( ( 1st `  ( 1st `  t
) )  =  A  /\  ( 2nd `  ( 1st `  t ) )  =  ( p ` 
1 )  /\  ( 2nd `  t )  =  B ) ) )
2014, 193anbi13d 1301 . . . . . 6  |-  ( ( a  =  A  /\  b  =  B )  ->  ( ( f ( a ( V SPathOn  E
) b ) p  /\  ( # `  f
)  =  2  /\  ( ( 1st `  ( 1st `  t ) )  =  a  /\  ( 2nd `  ( 1st `  t
) )  =  ( p `  1 )  /\  ( 2nd `  t
)  =  b ) )  <->  ( f ( A ( V SPathOn  E
) B ) p  /\  ( # `  f
)  =  2  /\  ( ( 1st `  ( 1st `  t ) )  =  A  /\  ( 2nd `  ( 1st `  t
) )  =  ( p `  1 )  /\  ( 2nd `  t
)  =  B ) ) ) )
21202exbidv 1692 . . . . 5  |-  ( ( a  =  A  /\  b  =  B )  ->  ( E. f E. p ( f ( a ( V SPathOn  E
) b ) p  /\  ( # `  f
)  =  2  /\  ( ( 1st `  ( 1st `  t ) )  =  a  /\  ( 2nd `  ( 1st `  t
) )  =  ( p `  1 )  /\  ( 2nd `  t
)  =  b ) )  <->  E. f E. p
( f ( A ( V SPathOn  E ) B ) p  /\  ( # `  f )  =  2  /\  (
( 1st `  ( 1st `  t ) )  =  A  /\  ( 2nd `  ( 1st `  t
) )  =  ( p `  1 )  /\  ( 2nd `  t
)  =  B ) ) ) )
2221rabbidv 3105 . . . 4  |-  ( ( a  =  A  /\  b  =  B )  ->  { t  e.  ( ( V  X.  V
)  X.  V )  |  E. f E. p ( f ( a ( V SPathOn  E
) b ) p  /\  ( # `  f
)  =  2  /\  ( ( 1st `  ( 1st `  t ) )  =  a  /\  ( 2nd `  ( 1st `  t
) )  =  ( p `  1 )  /\  ( 2nd `  t
)  =  b ) ) }  =  {
t  e.  ( ( V  X.  V )  X.  V )  |  E. f E. p
( f ( A ( V SPathOn  E ) B ) p  /\  ( # `  f )  =  2  /\  (
( 1st `  ( 1st `  t ) )  =  A  /\  ( 2nd `  ( 1st `  t
) )  =  ( p `  1 )  /\  ( 2nd `  t
)  =  B ) ) } )
23 eqid 2467 . . . 4  |-  ( a  e.  V ,  b  e.  V  |->  { t  e.  ( ( V  X.  V )  X.  V )  |  E. f E. p ( f ( a ( V SPathOn  E ) b ) p  /\  ( # `  f )  =  2  /\  ( ( 1st `  ( 1st `  t
) )  =  a  /\  ( 2nd `  ( 1st `  t ) )  =  ( p ` 
1 )  /\  ( 2nd `  t )  =  b ) ) } )  =  ( a  e.  V ,  b  e.  V  |->  { t  e.  ( ( V  X.  V )  X.  V )  |  E. f E. p ( f ( a ( V SPathOn  E ) b ) p  /\  ( # `  f )  =  2  /\  ( ( 1st `  ( 1st `  t
) )  =  a  /\  ( 2nd `  ( 1st `  t ) )  =  ( p ` 
1 )  /\  ( 2nd `  t )  =  b ) ) } )
2422, 23ovmpt2ga 6414 . . 3  |-  ( ( A  e.  V  /\  B  e.  V  /\  { t  e.  ( ( V  X.  V )  X.  V )  |  E. f E. p
( f ( A ( V SPathOn  E ) B ) p  /\  ( # `  f )  =  2  /\  (
( 1st `  ( 1st `  t ) )  =  A  /\  ( 2nd `  ( 1st `  t
) )  =  ( p `  1 )  /\  ( 2nd `  t
)  =  B ) ) }  e.  _V )  ->  ( A ( a  e.  V , 
b  e.  V  |->  { t  e.  ( ( V  X.  V )  X.  V )  |  E. f E. p
( f ( a ( V SPathOn  E )
b ) p  /\  ( # `  f )  =  2  /\  (
( 1st `  ( 1st `  t ) )  =  a  /\  ( 2nd `  ( 1st `  t
) )  =  ( p `  1 )  /\  ( 2nd `  t
)  =  b ) ) } ) B )  =  { t  e.  ( ( V  X.  V )  X.  V )  |  E. f E. p ( f ( A ( V SPathOn  E ) B ) p  /\  ( # `  f )  =  2  /\  ( ( 1st `  ( 1st `  t
) )  =  A  /\  ( 2nd `  ( 1st `  t ) )  =  ( p ` 
1 )  /\  ( 2nd `  t )  =  B ) ) } )
254, 5, 12, 24syl3anc 1228 . 2  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  ( A  e.  V  /\  B  e.  V ) )  -> 
( A ( a  e.  V ,  b  e.  V  |->  { t  e.  ( ( V  X.  V )  X.  V )  |  E. f E. p ( f ( a ( V SPathOn  E ) b ) p  /\  ( # `  f )  =  2  /\  ( ( 1st `  ( 1st `  t
) )  =  a  /\  ( 2nd `  ( 1st `  t ) )  =  ( p ` 
1 )  /\  ( 2nd `  t )  =  b ) ) } ) B )  =  { t  e.  ( ( V  X.  V
)  X.  V )  |  E. f E. p ( f ( A ( V SPathOn  E
) B ) p  /\  ( # `  f
)  =  2  /\  ( ( 1st `  ( 1st `  t ) )  =  A  /\  ( 2nd `  ( 1st `  t
) )  =  ( p `  1 )  /\  ( 2nd `  t
)  =  B ) ) } )
263, 25eqtrd 2508 1  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  ( A  e.  V  /\  B  e.  V ) )  -> 
( A ( V 2SPathOnOt  E ) B )  =  { t  e.  ( ( V  X.  V )  X.  V
)  |  E. f E. p ( f ( A ( V SPathOn  E
) B ) p  /\  ( # `  f
)  =  2  /\  ( ( 1st `  ( 1st `  t ) )  =  A  /\  ( 2nd `  ( 1st `  t
) )  =  ( p `  1 )  /\  ( 2nd `  t
)  =  B ) ) } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1379   E.wex 1596    e. wcel 1767   {crab 2818   _Vcvv 3113   class class class wbr 4447    X. cxp 4997   ` cfv 5586  (class class class)co 6282    |-> cmpt2 6284   1stc1st 6779   2ndc2nd 6780   1c1 9489   2c2 10581   #chash 12369   SPathOn cspthon 24181   2SPathOnOt c2pthonot 24533
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-1st 6781  df-2nd 6782  df-2spthonot 24536
This theorem is referenced by:  el2spthonot  24546  el2spthonot0  24547  2spthonot3v  24552  2pthwlkonot  24561  2spotfi  24568  2spotdisj  24738
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